= Solution
The region $D$ is compact and positively invariant and contains the unique fixed point. If that fixed point is unstable, choose an initial point in $D$ which is not on its stable set. Its nonempty compact $\omega$-limit set cannot be the fixed point. The <Poincaré-Bendixson theorem> therefore supplies a periodic orbit. Hence a sufficient parameter condition, and the one selected by the preceding analysis, is
$$
\boxed{0<a<\frac18,
\qquad
\frac{1-2a-\sqrt{1-8a}}2<b^2<
\frac{1-2a+\sqrt{1-8a}}2.}
$$
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